Lemma 2.2 (Section 3.1 of [1]})  Define
\( \phantom{\,.}\hspace{0.83328pt}\overline{\hspace{-0.83328pt}D\hspace{-0.83328pt}}\hspace{0.83328pt}_n(z) &= \sqrt{z} \bar{s}_n(z) \Big ( \beta _n \sqrt{z} \bar{s}_n(z) - \frac{1-\beta _n}{\sqrt{z}}\Big ) \nonumber \\&= -\frac{1+\beta _n - z + \sqrt{(1+\beta _n-z)^2 - 4 \beta _n}}{2 \beta _n} \,. \) 
